Once the unitary matrices \(U\)  and \(V\)  (or just their last \(r\)  columns, \(U_2\)  and \(V_2\) ) and the matrix \(\widehat{T}\)  in (REF ) are computed, Theorem  yields an efficient and stable algorithm for computing the strongly minimal linear realization \(\lbrace \widehat{A}_s (\lambda ), \widehat{B}_s (\lambda ), \widehat{C}_s (\lambda ) , \widehat{D}_s (\lambda ) \rbrace \)  of \(P(\lambda )\) . An expensive method for computing \(U\) , \(V\)  and \(\widehat{T}\)  is to compute the SVD of \(T\) , in economic version if just \(U_2\)  and \(V_2\)  are required. A cheaper method is to use the complete orthogonal decomposition in [1]}, which amounts to compute two QR factorizations. The block Hankel structure of \(T\)  (which by flipping the order of the block rows becomes block Toeplitz) allows us to use the very fast and stable method in [2]}, which makes all the computations on \(m\times n\)  submatrices. The method in [2]} has the additional advantage that if \(r_d = \mathrm {rank} (P_d)\) , then the rows of \(U^* (1:(m-r_d) , 1:m)\)  and the columns of \(V(1:n, 1 :(n-r_d))\)  (where we used MATLAB's notation) of the computed \(U\)  and \(V\)  are, respectively, unitary bases of the left and right nullspaces of \(P_d\) . Recall that these subspaces are precisely the left and right eigenspaces associated to the infinite eigenvalue of \(P(\lambda )\)  when \(P(\lambda )\)  is regular.
